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1: 18.40 Methods of Computation

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§18.40(ii) The Classical Moment Problem

►The problem of moments is simply stated and the early work of Stieltjes, Markov, and Chebyshev on this problem was the origin of the understanding of the importance of both continued fractions and OP’s in many areas of analysis. Given the power moments, ${\mu }_n={\int }_a^b{x}^{n}d\mu (x)$, $n=0,1,2,\mathrm{\dots }$, can these be used to find a unique $\mu (x)$, a non-decreasing, real, function of $x$, in the case that the moment problem is determined? Should a unique solution not exist the moment problem is then indeterminant. … ►A simple set of choices is spelled out in Gordon (1968) which gives a numerically stable algorithm for direct computation of the recursion coefficients in terms of the moments, followed by construction of the J-matrix and quadrature weights and abscissas, and we will follow this approach: Let $N$ be a positive integer and define … ►Having now directly connected computation of the quadrature abscissas and weights to the moments, what follows uses these for a Stieltjes–Perron inversion to regain $w(x)$. …

2: 18.2 General Orthogonal Polynomials

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§18.2(ix) Moments

►The moments for an orthogonality measure $d\mu (x)$ are the numbers …The monic OP’s $p_n(x)$ with respect to the measure $d\mu (x)$ can be expressed in terms of the moments by …Alternatives for numerical calculation of the recursion coefficients in terms of the moments are discussed in these references, and in §18.40(ii). … ►with moment ${\mu }_0$ defined in (18.2.26). …

3: Bibliography D

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Plancherel-Rotach asymptotic expansion for some polynomials from indeterminate moment problems. Constr. Approx. 40 (1), pp. 61–104.

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External Links:

ISSN 0176-4276, Document, Link, MathReview (Francisco Marcellán), ZentralBlatt

Referenced by:

§2.9(iii), §2.9(iii).

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Functions with given moments and weight functions for orthogonal polynomials. Rocky Mountain J. Math. 23, pp. 87–104.

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External Links:

ISSN 0035-7596, MathReview (Olav Njåstad), ZentralBlatt

Referenced by:

§18.34(ii).

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4: Bibliography L

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Moment-theory investigations of photoabsorption and dispersion profiles in atoms and ions. Phys. Rev. A 14, pp. 1042–1056.

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External Links:

Document, Link

Referenced by:

§18.40(ii).

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Moment integrals of powers of Airy functions. Z. Angew. Math. Phys. 44 (5), pp. 891–908.

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External Links:

ISSN 0044-2275, Document, MathReview (B. M. Agrawal), ZentralBlatt

Referenced by:

(9.11.15), (9.11.18), §9.11(v), §9.11(v).

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5: 18.34 Bessel Polynomials

… ►The full system satisfies orthogonality with respect to a (not positive definite) moment functional; see Evans et al. (1993, (2.7)) for the simple expression of the moments ${\mu }_n$. …

6: Bibliography W

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The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A, pp. 32–52.

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External Links:

FullText, Document, ZentralBlatt

Referenced by:

§35.3(i), §35.3(ii).

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7: Bibliography J

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Asymptotic behavior of the continued fraction coefficients of a class of Stieltjes transforms including the Binet function. In Orthogonal functions, moment theory, and continued fractions (Campinas, 1996), Lecture Notes in Pure and Appl. Math., Vol. 199, pp. 257–274.

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External Links:

MathReview (Olav Njåstad), ZentralBlatt

Referenced by:

§5.10, §5.10.

Encodings:

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8: Bibliography

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The classical moment problem and some related questions in analysis. Classics in Applied Mathematics, Vol. 82, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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Notes:

Reprint of the 1965 edition [ 0184042], Translated by N. Kemmer, With a foreword by H. J. Landau

External Links:

ISBN 978-1-611976-38-0, MathReview Entry

Referenced by:

§18.40(ii).

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9: Bibliography G

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Matrices, moments and quadrature with applications. Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ.

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External Links:

ISBN 978-0-691-14341-5, MathReview (G. A. Evans)

Referenced by:

§18.2(ix), §18.40(ii).

Encodings:

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10: Bibliography B

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Moment equations for probability distributions in classical and quantum mechanics. Phys. Rev. A 58 (3), pp. 1799–1809.

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External Links:

Document

Referenced by:

§24.18.

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